# The Structure of the Class of Maximum Tsallis–Havrda–Chavát Entropy Copulas

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**I**=$[0,1],$ which maximizes the THC entropy ${T}_{2}(f),$ given by

## 3. Results

**I**, the inverse ${F}_{i}$ of ${F}_{i}^{-1}$ is an absolutely continuous distribution function on

**I**. We find necessary and sufficient conditions on the functions ${F}_{1}$ and ${F}_{2}$ (in terms of the derivatives of their inverses) for the ME copula in Equation (6) to be a RU copula. These conditions also insure that the functions ${F}_{1}$ and ${F}_{2}$ yield a proper joint distribution function in (5).

**Theorem**

**1.**

- i
- Every ME copula has the functional form of a RU copula.
- ii
- Let ${F}_{1}^{-1}(u)$ and ${F}_{2}^{-1}(v)$ be inverse distribution functions on
**I**, and set $A=\{u\in \mathbf{I}:\frac{d}{du}{F}_{1}^{-1}(u)\phantom{\rule{0.166667em}{0ex}}exists\},$ $B=\{v\in \mathbf{I}:\frac{d}{dv}{F}_{2}^{-1}(v)\phantom{\rule{0.166667em}{0ex}}exists\},$ ${m}_{1}=inf\{\frac{d}{du}{F}_{1}^{-1}(u):u\in A\},$ ${M}_{1}=sup\{\frac{d}{du}{F}_{1}^{-1}(u):u\in A\},$ ${m}_{2}=inf\{\frac{d}{dv}{F}_{2}^{-1}(v):v\in B\},$ ${M}_{2}=sup\{\frac{d}{dv}{F}_{2}^{-1}(v):v\in B\}.$ Then the function $C(u,v)$ in Equation (6) is a copula-in fact a RU copula-if and only if $max\{({M}_{1}-1)({M}_{2}-1),({m}_{1}-1)({m}_{2}-1)\}\le 1.$

**Proof.**

- i
- Equation (6) is equivalent to$$C(u,v)=uv+[u-{F}_{1}^{-1}(u)][{F}_{2}^{-1}(v)-v],$$
- ii
- Since ${f}^{\prime}(u)=1-\frac{d}{du}{F}_{1}^{-1}(u)$ and ${g}^{\prime}(v)=\frac{d}{dv}{F}_{2}^{-1}(v)-1,$ $\alpha =1-{M}_{1}=inf\{{f}^{\prime}(u):u\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}A\},$ $\beta =1-{m}_{1}=sup\{{f}^{\prime}(u):u\in A\},$ $\gamma ={m}_{2}-1=inf\{{g}^{\prime}(v):v\in B\},$ and $\delta ={M}_{2}-1=sup\{{g}^{\prime}(v):v\in B\}.$ The condition $min\{\alpha \delta ,\beta \gamma \}\ge -1$ given in Theorem 2.3 [5] is equivalent to $max\{({M}_{1}-1)({M}_{2}-1),({m}_{1}-1)({m}_{2}-1)\}\le 1,$ and the conclusion follows.

**Example**

**1.**

**Example**

**2.**

**Theorem**

**2.**

**Proof.**

**I**. Clearly, from Theorem 2.3 in [5], ${G}_{1}(0)={G}_{2}(0)=0$ and ${G}_{1}(1)={G}_{2}(1)=1,$ so it suffices to show that ${G}_{1}^{\prime}(u)=1-k{f}^{\prime}(u)\ge 0$ and ${G}_{2}^{\prime}(v)=\frac{1}{k}{g}^{\prime}(v)+1\ge 0$ for some $k>0.$ Since $min\{\alpha \delta ,\beta \gamma \}\ge -1$ from Theorem 2.3 in [5], both $\alpha \delta \ge -1$ and $\beta \gamma \ge -1.$ But $\beta >0$ and $\gamma <0$ imply that $0<-\gamma \le \frac{1}{\beta},$ hence there exists a $k>0$ such that $-\gamma \le k\le \frac{1}{\beta}.$ Thus ${f}^{\prime}(u)\le \beta $ implies that $k{f}^{\prime}(u)\le k\beta \le 1,$ and hence ${G}_{1}^{\prime}(u)\ge 0.$ Similarly ${g}^{\prime}(v)\ge \gamma $ implies that $\frac{1}{k}{g}^{\prime}(v)\ge \frac{1}{k}\gamma \ge -1,$ and hence ${G}_{2}^{\prime}(v)\ge 0.$ Thus ${G}_{1}(u)$ and ${G}_{2}(v)$ are inverse distribution functions on

**I**. A similar result follows from $\alpha \delta \ge -1$ setting ${G}_{1}(u)=kf(u)+u$ and ${G}_{2}(v)=v-\frac{1}{k}g(v).$ ☐

**Example**

**3.**

**Example**

**4.**

**I**with $a\in [-1,1].$ Then the copula C in Equation (6) with maximum entropy is $C(u,v)=uv+{a}^{3}uv(1-u)(1-v),$ a Farlie–Gumbel–Morgenstern copula with parameter $\theta ={a}^{3}.$ Note that ${m}_{1}=1-{a}^{2},{M}_{1}=1+{a}^{2},$ ${m}_{2}=min\{1-a,1+a\},$ and ${M}_{2}=max\{1-a,1+a\}.$ The joint distribution F given by Equation (5) has entropy ${T}_{2}(F)=2-\frac{1}{2{a}^{2}}ln\left(\frac{1+{a}^{2}}{{(1+a)}^{1-a}{(1-a)}^{1+a}}\right)$ for $a\ne 0,$ and ${T}_{2}(F)=0$ for $a=0$, where ${F}_{1}(x)=\frac{1+{a}^{2}-\sqrt{1+2{a}^{2}+{a}^{4}-4{a}^{2}x}}{2{a}^{2}},$ ${f}_{1}(x)=\frac{1}{\sqrt{1+2{a}^{2}+{a}^{4}-4{a}^{2}x}}$ and ${F}_{2}(y)=\frac{a-1+\sqrt{{(a-1)}^{2}+4ay}}{2a},$ ${f}_{2}(y)=\frac{1}{\sqrt{1-2a+{a}^{2}+4ay}}.$

**Example**

**5.**

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BGS | Boltzmann–Gibbs–Shannon |

THC | Tsallis–Havrda–Chavát |

ME | maximum entropy |

RU | Rodríguez-Lallena and Úbeda-Flores |

FBST | Full Bayesian Significance Test |

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**Table 1.**Evidence against ${H}_{0}:\theta =0,$ prior density on θ $\propto {(\frac{1+\theta}{2})}^{a-1}{(\frac{1-\theta}{2})}^{b-1}.$

Setting | a | b | Ev |
---|---|---|---|

(i) | 1 | 1 | 0.02686 |

(ii) | 2 | 2 | 0.05580 |

(iii) | 2 | 4 | 0.01702 |

(iv) | 4 | 2 | 0.28670 |

(v) | 2 | 10 | 0.00050 |

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**MDPI and ACS Style**

García, J.E.; González-López, V.A.; Nelsen, R.B.
The Structure of the Class of Maximum Tsallis–Havrda–Chavát Entropy Copulas. *Entropy* **2016**, *18*, 264.
https://doi.org/10.3390/e18070264

**AMA Style**

García JE, González-López VA, Nelsen RB.
The Structure of the Class of Maximum Tsallis–Havrda–Chavát Entropy Copulas. *Entropy*. 2016; 18(7):264.
https://doi.org/10.3390/e18070264

**Chicago/Turabian Style**

García, Jesús E., Verónica A. González-López, and Roger B. Nelsen.
2016. "The Structure of the Class of Maximum Tsallis–Havrda–Chavát Entropy Copulas" *Entropy* 18, no. 7: 264.
https://doi.org/10.3390/e18070264